Ind. Eng. Chem. Fundam. 1981, 20, 101-102
101
Mechanism of Drag Reduction in Turbulent Pipe Flow by the Addition of Fibers On the concept of suppression of viscous eddies near the pipe wall of turbulent pipe flow, an expression describing drag reduction by fibers as a function of concentration and aspect ratio is derived and shown to agree with experimental results.
The reduction of drag in turbulent pipe flow by the addition of fibers has many practical applications. Recent studies on the subject have been done by Kerekes and Douglas (1972),Hoyt (1972),Vaseleski and Metzner (19741, Radin et al. (1975), and Kale and Metzner (1976). It is generally believed that drag reduction by fibers is due to an interaction between fibers and the eddies of turbulent pipe flow. Kale and Metzner (1976) established the origin of drag reduction in the region of the flow very close to pipe wall where eddies are mainly of dissipative type and form the viscous zone of pipe flow turbulence. It thus appears that an interaction between viscous eddies and fibers could be a possible cause for drag reduction in turbulent pipe flow. In turbulent flow conditions, eddies are constantly stretched by the action of velocity fluctuations. Fibers in similar situations have been found by Mewis and Metzner (1970) to exhibit very high resistance to extensional deformations. It appears reasonable to hold the view that the viscous eddies of the flow are suppressed by fibers. As these eddies are the characteristic eddies of the flow, suppression of them can modify the entire turbulence in a direction that could lead to a reduced level of radial momentum transfer to give drag reduction. In this communication,the influence of fibers on viscous eddies is expressed through extensional viscosity to obtain an expression for drag reduction in the turbulent pipe flow of fiber suspensions. The fractional drag reduction can be defined by
DR=i-($) u
=
up
in which friction factor f is given by f = 2(uo/W* with this, eq 1 can be written as DR = 1 - ( u o , / u o ) ~
(2) (3)
Friction velocity uo can be expressed by the Kolmogoroff velocity scale U X (Davies, 1972) ux =
(4'14
(4)
Substituting friction velocity by u X , eq 3 can be expressed by DR = 1 - ( t P ~ , / t v ) ' I 2 (5) The eddy length corresponding to uh is ( v ~ / c ) ' /and ~ , for a constant eddy length it can be shown that tp/t
=
(V/VJ3
With this, eq 5 can be written as DR = 1 - V/V,
(6) (7)
In eq 7, vp is the eddy viscosity of viscous eddies and as such should be given by extensional viscosity. In the absence of a relevant expression for vp it is assumed that at a fixed Reynolds number v/v is a function of concentration and aspect ratio of fibers. kriting v / v p = +(4,r) DR = 1 - +(4,r) (8) 0196-4313181 11020-0101$01.0010
In this, DR varies from zero to a maximum value, DR,,. Therefore, +($,r) should decrease from 1to a value smaller than 1 with the increase of 4 and r. A function given in eq 9 is found to satisfy these conditions.
W , r ) = exP(-C(4/4max)') (9) P in eq 9 is a function of r. Substituting +(4,r) into eq 8 and then converting the resulting equation into a log form gives
Constant C is given by C = -In (1- DR,,)
(11)
In terms of friction factor eq 10 becomes log ln
V / f J = P log 4 - P log @ma, + log C (12)
To check on eq 12 and to determine and C, the results of Kerekes and Douglas (1972), who measured friction losses for nylon fibers suspended in water, are treated by plotting In ( f / f p ) vs. 4 in a log plot as shown in Figure 1. In following straight lines these results show good agreement with eq 12. The slope of the lines representing P correlates with aspect ratio to give = 81r-1.1
(13)
As can be seen in Figure 1, the lines at different aspect ratio can be extended to meet at a point and this gives C and 4, 1.7 and 0.10, respectively. With C = 1.7, DR, from eq 11is 0.817. Kerekes and Douglas (1972), however, found maximum drag reduction to depend on concentration and aspect ratio with values smaller than 0.817. This behavior can be seen considering the fact that fibers in turbulent flow suffer collisions costing the flow some energy that appears as drag and with the increase of concentration and aspect ratio the rate of collisions increases causing drag increase. In addition to this effect, drag increase also occurs through the increase of suspension viscosity with concentration and aspect ratio. At some point of these variables drag increase supersedes drag reduction before it reaches the predicted maximum value. Taking the antilog in eq 12, friction factor f, can be expressed by f p
=f
C(4/4max)B)
(14)
from which it is evident that the relationship between f p and Reynolds number at constant 4 and r should be the same as that between f and Re. But experimental results (Radin et al., 1975) show differently; the decrease of f, at higher Reynolds number is found to be slower compared with f. This behavior can also be explained attributing the cause to the increase of collision rate among fibers with Reynolds number. Apart from this, the proposed mechanism of drag reduction by fibers in turbulent pipe flow is found to be valid over the range of concentration and aspect ratio values where energy losses through interfiber collisions and sus0 1981 American Chemical Society
Ind. Eng. Chem. Fundam. 1981, 20,
102
102-104
L = length of fiber r = aspect ratio = L / d C' = average velocity vX = eddy velocity vo = friction velocity Greek L e t t e r s /3 = function of aspect ratio t
= energy dissipation rate
X = eddy scale u = kinematic viscosity u p = eddy viscosity = vX,hp 4 = concentration of fiber in volume fraction
Subscripts
max = maximum p = fiber
Literature Cited 33,L.0 301
-
-
--
09'
~
- 010
@ of In (f/fp) vs. 9, comparison of eq 12 with the results of Kerekes and Douglas (1972). F i g u r e 1. Plot
pension viscosity are not high enough to reverse drag reduction to drag increase. Nomenclature C = constant in Eq 9 DR = fractional drag reduction d = diameter of fiber f = friction factor
Davies, J. T. "Turbulence Phenomena", Academic Press: New York, 1972; p 27. Hoyt, J. W. "Naval Undersea Center Rept. T.P. 299", San Diego, Calif., 1972. Kale, D. D.; Metzner, A. B. AIChE J . 1976, 22, 669. Kerekes, R. J. E.; Douglas, W. J. M. Can. J . Chem. Eng. 1972, 50, 228. Mewis, J.; Metzner, A. B. J. N u M M c h . 1974, 62, 593. Radin, I.; J. L.; Patterson, G. K. AIChE J. 1975, 21, 358. Vaseleski. R. C.:Metzner, A. 6 . AIChE J . 1974, 20, 301.
Department of Chemical Engineering Polytechnic of Wales Pontypridd, Mid- Glamorgan CF37 1DL South- Wales, United Kingdom
M. S. Doulah
Received for review April 15, 1980 Accepted September 26, 1980
Residence-Time Distributions for Systems Having Many Connections with Their Environments The difference reported by Ritchie and Tobgy (1978) between their expression for the residence-time density function for a closed system and that previously established by Buffham and Kropholler (1970) is because Riche and Tobgy's analysis is not consistent with their definitions. The agreement they found between their equation and that of Treleaven and Tobgy (1971) for systems having two inlets and one exit is fortuitous. Functions relating the time spent in the system and the place whence material leaves to the location at which it entered may be defined in different ways. Correct expressions for the residence-timedensity function are given for several definitions and collected into a table for easy comparison.
Introduction In a recent paper, Ritchie and Tobgy (1978) have presented a residence-time analysis for systems having many connections with their environments. They stress that their relation describing the overall residence-time frequency function differs from that previously published by us (Buffham and Kropholler, 1970). Our purpose here is to explain the difference between the two formulations and to indicate the flaws in Ritchie and Tobgy's treatment, lest it be thought that the method is fundamentally unsound. The system under consideration has several connections with its environment through which material passes by bulk flow so that the connections may be classified unambiguously as inlets and outlets. In the steady state, the residence-time distribution is the distribution of ages of material leaving the system or, equivalently, the distribution of life expectancies of material entering the system. Our analysis was in terms of the concentration responses at the outlets to concentration forcing at the inlets. Ritchie 0196-43 13/81/1020-0102$01 .OO/O
and Tobgy's analysis is in terms of frequency functions. The treatment that follows uses frequency functions. We have discussed a somewhat more general case elsewhere (Buffham and Kropholler, 1973). Frequency Analysis Our principal concern is the discrepancy between Ritchie and Tobgy's (1978) eq 17 and eq 9 of our paper (Buffham and Kropholler, 1970). We shall refer to these equations as eq RT17 and eq BK9 and refer similarly to other equations from these papers. Ritchie and Tobgy express eq RT17 in terms of a function fij(t) defined by (definition 0 : f , ( t ) d t = the fraction of the ith inlet stream which leaves via the j t h outlet stream with residence time between t and t + dt. Equations RT17 and BK9 differ, partly because eq RT17 does not follow from definition I and partly because f,,(t) in eq RT17 is not the same as fij(t) in eq BK9. First, we shall develop the correct expression for the residencetime density from definition I; then we shall give frequency-function definitions which lead to residence-time 0 1981 American
Chemical Society
